Definition 39.4.1. Let $S$ be a scheme.

A

*group scheme over $S$*is a pair $(G, m)$, where $G$ is a scheme over $S$ and $m : G \times _ S G \to G$ is a morphism of schemes over $S$ with the following property: For every scheme $T$ over $S$ the pair $(G(T), m)$ is a group.A

*morphism $\psi : (G, m) \to (G', m')$ of group schemes over $S$*is a morphism $\psi : G \to G'$ of schemes over $S$ such that for every $T/S$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups.

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